Twenty years have gone by since the [1]} paper in which Rob Engle provided a framework of generalization to the ARCH [2]} and GARCH [3]} modeling of the conditional variance of financial returns. Following the work in [4]}, and the overall remarks in [5]}, it was recognized that the availability of ultra–high frequency data had opened the venue to widening the range of the analysis of market activity.
New financial time series became readily available: [6]} started a stream of literature addressing the properties of measuring volatility from intradaily data and using it as an alternative target to the squared returns (a very noisy measure of return variance); for a review of various realized volatility measures, see [7]}; for the care to be exerted in working with ultra–high frequency data, see [8]}. While in the GARCH literature measurement and forecasting were steps performed simultaneously, the development of volatility measures left the question open as of what model would be the most appropriate to derive forecasts. Volatility being non–negatively valued and persistent lends these measures to be represented in some autoregressive fashion; as expressed by market trading activity volatility is accompanied by other indicators which exhibit the stylized facts of volatility clustering: next to realized volatility and durations, we can mention the volume, range, number of trades, bid–ask spread. In [1]}, a new class of models was suggested for non–negative valued processes, called Multiplicative Error Models (MEM), obtained by multiplicatively combining the conditional expectation of a process (deterministically dependent upon an information set at a previous time period) with a random disturbance representing unpredictable news. This approach is different in spirit from the linear and additive modeling suggested by [10]} in his Heterogeneous Autoregressive (HAR) model for realized variances, which has the merit of recognizing that long–memory features, often cited in the profile of realized volatility especially, can be mimicked upon suitable aggregation of the lagged variances [11]}.
